Weighted Vertex Cover Problem Research Articles

Polynomial-Time Approximation Schemes for Geometric Intersection Graphs

A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomial-time approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weight) and for the minimum weight vertex cover problem in disk graphs. These are the first known PTASs for $\mathcal{NP}$-hard optimization problems on disk graphs. They are based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible. The PTASs for disk graphs represent a common generalization of previous results for planar graphs and unit disk graphs. They can be extended to intersection graphs of other "disk-like" geometric objects (such as squares or regular polygons), also in higher dimensions.

An Ant Colony Optimization Algorithm for the Minimum Weight Vertex Cover Problem

Given an undirected graph and a weighting function defined on the vertex set, the minimum weight vertex cover problem is to find a vertex subset whose total weight is minimum subject to the premise that the selected vertices cover all edges in the graph. In this paper, we introduce a meta-heuristic based upon the Ant Colony Optimization (ACO) approach, to find approximate solutions to the minimum weight vertex cover problem. In the literature, the ACO approach has been successfully applied to several well-known combinatorial optimization problems whose solutions might be in the form of paths on the associated graphs. A solution to the minimum weight vertex cover problem however needs not to constitute a path. The ACO algorithm proposed in this paper incorporates several new features so as to select vertices out of the vertex set whereas the total weight can be minimized as much as possible. Computational experiments are designed and conducted to study the performance of our proposed approach. Numerical results evince that the ACO algorithm demonstrates significant effectiveness and robustness in solving the minimum weight vertex cover problem.

Algorithms column

In this column I give a slightly simpler proof of an old result by Nemhauser and Trotter [10]. It would be interesting to see for which other problems such results hold. One of the widely studied problems in Combinatorial Optimization is the Weighted Vertex Cover problem. Given a graph G = (V, E) with a weight function defined on the vertices, a vertex cover is a subset of vertices S such that for each edge e = (u, v) either u E S or v 6 S. The minimum weight vertex cover problem asks for a vertex cover of minimum total weight. This problem has been the subject of many papers, since the problem is NP-hard, but can be solved optimally for bipartite graphs by a reduction to network flows. There are many papers addressing the issue of obtaining polynomial time approximation algorithms [1, 2, 3, 4, 5, 8, 7]. One of the first approaches (in fact this also implies a polynomial time 2 approximation), is by Nemhauser and Trotter [10]. Consider the following simple Integer Program (IP) for the Weighted Vertex Cover Problem. Here Xu refers to a binary indicator variable, and its value is 1 if and only if vertex u is in the cover.

A 2-approximation algorithm for the minimum weight edge dominating set problem

We present a polynomial-time algorithm approximating the minimum weight edge dominating set problem within a factor of 2. It has been known that the problem is NP-hard but, when edge weights are uniform (so that the smaller the better), it can be efficiently approximated within a factor of 2. When general weights were allowed, however, very little had been known about its approximability, and only very recently was it shown to be approximable within a factor of 2 1 10 by reducing to the edge cover problem via LP relaxation. In this paper we extend the approach given therein, by studying more carefully polyhedral structures of the problem, and obtain an improved approximation bound as a result. While the problem considered is as hard to approximate as the weighted vertex cover problem is, the best approximation (constant) factor known for vertex cover is 2 even for the unweighted case, and has not been improved in a long time, indicating that improving our result would be quite difficult.

A 2\frac{1}{10}-Approximation Algorithm for a Generalization of the Weighted Edge-Dominating Set Problem.

We study the approximability of the weighted edge-dominating set problem. Although even the unweighted case is NP-Complete, in this case a solution of size at most twice the minimum can be efficiently computed due to its close relationship with minimum maximal matching; however, in the weighted case such a nice relationship is not known to exist. In this paper, after showing that weighted edge domination is as hard to approximate as the well studied weighted vertex cover problem, we consider a natural strategy, reducing edge-dominating set to edge cover. Our main result is a simple $$2\frac{1}{{10}}$$ -approximation algorithm for the weighted edge-dominating set problem, improving the existing ratio, due to a simple reduction to weighted vertex cover, of 2r WVC, where r WVC is the approximation guarantee of any polynomial-time weighted vertex cover algorithm. The best value of r WVC currently stands at $$2 - \frac{{\log \log \left| V \right|}}{{2\log \left| V \right|}}$$ . Furthermore we establish that the factor of $$2\frac{1}{{10}}$$ is tight in the sense that it coincides with the integrality gap incurred by a natural linear programming relaxation of the problem.

A simple LP-free approximation algorithm for the minimum weight vertex cover problem

We present a simple LP-free (i.e., not requiring linear programming) approximation algorithm for the minimum weight vertex cover problem. Our new approximation algorithm does not need to solve a linear programming problem, nor such a formulation is needed to establish its approximation bound. The algorithm takes linear time with respect to the number of nodes and edges in the graph, and generates solutions that are within twice the weight of a minimum weight vertex cover. Both the algorithm and its proof of correctness are elegant and simple.

A parallel algorithm for the minimum weighted vertex cover problem

An original parallel algorithm is presented that stems from a suitable physical metaphor by considering each edge of the graph as a cell that is attracted by the corresponding vertices and finally moves towards the one of them that wins the competition.

Can Parallel Branch and Bound without Communication Be Effective?

The usual approach to parallel Branch and Bound is to employ dynamic exchange of subproblems during the search in order to ensure a uniform workload on all processors. It is a common assumption that static distribution of subproblems is “inherently inefficient.” A parallel Branch and Bound algorithm was implemented, which does not communicate during the search. A small number of subproblems are initially generated and distributed among the processors which then perform ordinary serial Branch and Bound. Since the value of the current best solution is not distributed during the search it is necessary to be able to provide a good initial solution. The algorithm has been applied to three combinatorial optimization problems; the quadratic assignment problem, the graph partitioning problem, and the weighted vertex cover problem. For all three problems, an average processor utilization of 0.70–0.85 was observed. Parallel Branch and Bound without communication is thus not inherently inefficient.

An experimental comparison of three heuristics for the WVCP

The modification of the greedy algorithm for the weighted vertex cover problem (WVCP) suggested by Clarkson yields an algorithm with better worst-case performance. The results presented in this paper indicate, however, that the generic greedy algorithm performs far better on the average than the modified algorithm. Even so, the generic greedy algorithm produces solutions which are on the average several percent above optimum, and which cannot be improved by additional runs due to the deterministic nature of the algorithm. The nondeterministic Simulated Annealing algorithm has therefore been specialized to the WVCP. The Simulated Annealing algorithm yields solutions of higher quality, even when only a moderate number of iterations are performed.

A linear-time approximation algorithm for the weighted vertex cover problem

A linear time approximation algorithm for the weighted set-covering problem is presented. For the special case of the weighted vertex cover problem it produces a solution of weight which is at most twice the weight of an optimal solution.